Question

If $z\ne 0$ is a complex number, then

• A. $\mathrm{Re}\left(z\right)=0\Rightarrow \mathrm{Im}\left(z^2\right)=0$
• B. $\mathrm{Re}\left(z^2\right)=0\Rightarrow \mathrm{Im}\left(z^2\right)=0$
• C. $\mathrm{Re}\left(z\right)=0\Rightarrow \mathrm{Re}\left(z^2\right)=0$
• D. None of these

Answer 1

The correct option is A

$\mathrm{Re}\left(z\right)=0\Rightarrow \mathrm{Im}\left(z^2\right)=0$

Explanation for the correct option.

Find the correct relation.

Let $z=x+iy$ its real part is $x$ and imaginary part is $y$.

Let us assume that $\mathrm{Re}\left(z\right)=0$ that is $x=0$.

Then, $z=iy$ and $z^2$ is given as:

$\begin{array}{rcl}{z}^2& =& {\left(iy\right)}^2\\ & =& i^2{y}^2\\ & =& -y^2\left[i^2=-1\right]\end{array}$

So the complex number $z^2$ is purely real because it has no imaginary part that is $\mathrm{Im}\left(z^2\right)=0$.

So if $\mathrm{Re}\left(z\right)=0$ then it implies that $\mathrm{Im}\left(z^2\right)=0$.

Hence, the correct option is A.